Sensitivity Analysis Jake Blanchard Fall 2010 Introduction Sensitivity

Sensitivity Analysis Jake Blanchard Fall 2010

Introduction �Sensitivity Analysis = the study of how uncertainty in the output of a model can be apportioned to different input parameters �Local sensitivity = focus on sensitivity at a particular set of input parameters, usually using gradients or partial derivatives �Global or domain-wide sensitivity = consider entire range of inputs

Typical Approach �Consider problem a Point Reactor Kinetics

Results �P(t) normalized to P 0 �Mean lifetime normalized to baseline value (0. 001 s) �t=3 s

Results �P(t) normalized to P 0 �Mean lifetime normalized to baseline value (0. 001 s) �t=0. 1 s

Putting all on one chart – t=0. 1 s

Putting all on one chart – t=3 s

Quantifying Sensitivity �To first order, our measure of sensitivity is the gradient of an output with respect to some particular input variable. �Suppose all variables are uncertain and �Then, if inputs are independent,

Quantifying Sensitivity �Most obvious calculation of sensitivity is �This is the slope of the curves we just looked at �We can normalize about some point (y 0)

Quantifying Sensitivity �This normalized sensitivity says nothing about the expected variation in the inputs. �If we are highly sensitive to a variable which varies little, it may not matter in the end �Normalize to input variances

Rewriting…

A Different Approach �Question: If we could eliminate the variation in a single input variable, how much would we reduce output variation? �Hold one input (Px) constant �Find output variance – V(Y|Px=px) �This will vary as we vary px �So now do this for a variety of values of px and find expected value E(V(Y|Px))

Now normalize �This ◦ ◦ is often called the importance measure, sensitivity index, correlation ratio, or first order effect

Variance-Based Methods �Assume �Choose each term such that it has a mean of 0 �Hence, f 0 is average of f(x)

Variance Methods �Since terms are orthogonal, we can square everything and integrate over our domain

Variance Methods �Si is first order (or main) effect of xi �Sij is second order index. It measures effect of pure interaction between any pair of output variables �Other values of S are higher order indices �“Typical” sensitivity analysis just addresses first order effects �An “exhaustive” sensitivity analysis would address other indices as well

Suppose k=4 � 1=S 1+S 2+S 3+S 4+S 12+S 13+S 14+S 23+S 24 +S 34+S 123+S 124+S 134+S 234+S 1234 �Total # of terms is 4+6+4+1=15=24 -1